Difference between revisions of "Ihara zeta function"
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(Created page with "Let $X$ be a finite graph. The Ihara zeta function is given by the formula $$\zeta_X(t) = \displaystyle\prod_{[C]} \dfrac{1}{1-t^{|C|}},$$ where $[C]$ is the set of equiva...") |
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Let $X$ be a finite [[graph]]. The Ihara zeta function is given by the formula | Let $X$ be a finite [[graph]]. The Ihara zeta function is given by the formula | ||
$$\zeta_X(t) = \displaystyle\prod_{[C]} \dfrac{1}{1-t^{|C|}},$$ | $$\zeta_X(t) = \displaystyle\prod_{[C]} \dfrac{1}{1-t^{|C|}},$$ | ||
| − | where $[C]$ is the set of equivalence classes of primitive closed paths $C$ in $X$ and $|C|$ denotes the length of $C$. This formula is an analogue of the Euler product representation of the [[Riemann zeta function]]. | + | where $[C]$ is the set of equivalence classes of primitive closed paths $C$ in $X$ and $|C|$ denotes the length of $C$. This formula is an analogue of the [[Euler product]] representation of the [[Riemann zeta function]]. |
=References= | =References= | ||
[http://math.tamu.edu/~grigorch/publications/zukiharanewbisk2.ps The Ihara zeta function of infinite graphs, the KNS spectral measure and integrable maps - Rostislav I. Grigorchuk, Andrzej Zuk] | [http://math.tamu.edu/~grigorch/publications/zukiharanewbisk2.ps The Ihara zeta function of infinite graphs, the KNS spectral measure and integrable maps - Rostislav I. Grigorchuk, Andrzej Zuk] | ||
Revision as of 00:42, 3 August 2014
Let $X$ be a finite graph. The Ihara zeta function is given by the formula $$\zeta_X(t) = \displaystyle\prod_{[C]} \dfrac{1}{1-t^{|C|}},$$ where $[C]$ is the set of equivalence classes of primitive closed paths $C$ in $X$ and $|C|$ denotes the length of $C$. This formula is an analogue of the Euler product representation of the Riemann zeta function.
